110 CHAPTER 4 The Approximate Scalar Potential: Properties and Shortcomings
Proposition 4.5.
Assume ~t constant in D. For all inner nodes
n, Mmn G 0
for all m ¢: n.
Proof.
Edge mn is flanked by two triangles T and T' (Fig. 4.10), and the
opposite angles add up to less than 180 ° , thanks to the circle property.
By the cotangent formula (3.23), - M m
n
is proportional to cot 0 + cot 0',
which is >0 if 0+0'<~. 0
Alas, this leaves many loose strands. First, obtuse angles at boundary
triangles (there is one in Fig. 4.8, triangle {8, 10, 11}). But if the objective
is to enforce the discrete maximum principle, only inner nodes are
involved, and anyway, one may add nodes at the boundary and, if
necessary, remesh (which is a local, inexpensive process). Second, and
more serious, the condition of uniformity of ~t is overly restrictive, and
although the cure is of the same kind (add nodes at discontinuity interfaces
to force acute angles), further research is needed in this direction.
.- ....... .-....
n ..........
i ........ M'
.......................................... • .
FIGURE 4.10. Proof of (5). M and M' are the mid-edges.
Proposition 4.5 can be proven in a different and instructive way (Fig.
4.10). By the cotangent formula, and the obvious angular relation of Fig.
4.10 (where C and C' are the circumcenters), one has
(5)
1
-
Mmn - -~
(~t(T) cot 0
+
~t(T') cot 0') = (h
~(T) +
h'
~t(T'))/ I mnl,
9Conventionally, the conditions reported by a meteorological station (temperature,
hygrometry, etc.) are supposed to hold in the whole "Thiessen polygon" around that station.
As explained in [CH], "Stations are always being added, deleted, moved, or temporarily
dropped from the network when they fail to report for short periods of time
(missing data)", hence the necessity to frequently solve the typical problem: having a
Voronoi-Delaunay mesh, add or delete a node, and recalculate the boundaries. Recursive
application of this procedure is the standard Watson-Bowyer algorithm for VD mesh
generation [Wa], much improved recently [SI] by making it resistant to roundoff errors.
Fine displays of VD meshes can be found in [We].
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